Understanding Gravitational Force and its Dependence on Distance

The question asks about how the force of attraction between two objects relates to the distance separating them. This relationship is described by Newton's Law of Universal Gravitation.

Newton's Law of Universal Gravitation

Sir Isaac Newton formulated the law that describes the gravitational force between any two objects with mass. The law states:

• The force of attraction between two point masses is directly proportional to the product of their masses.
• The force of attraction between two point masses is inversely proportional to the square of the distance between them.

Mathematically, the force \( F \) between two objects of masses \( M \) and \( m \) separated by a distance \( d \) is given by the formula:

In this formula:

• \( F \) represents the force of gravitational attraction.
• \( G \) is the universal gravitational constant, a value that is the same everywhere in the universe.
• \( M \) and \( m \) are the masses of the two objects.
• \( d \) is the distance between the centers of the two objects.

Analyzing the Inverse Proportionality

The question specifically asks about the inverse proportionality of the force with respect to the distance \( d \).

From the formula \( F = G \frac{Mm}{d^2} \), we can observe the relationship:

• The terms \( G \), \( M \), and \( m \) are specific constants for a given pair of objects. If these are kept constant, the force \( F \) depends only on \( d \).
• The relationship can be written as \( F \propto \frac{1}{d^2} \).

This expression \( F \propto \frac{1}{d^2} \) means that the force \( F \) is proportional to the reciprocal of \( d^2 \). In physics, when a quantity is proportional to the reciprocal of another quantity, it is said to be inversely proportional to that quantity.

Here, \( F \) is proportional to \( \frac{1}{d^2} \), so \( F \) is inversely proportional to \( d^2 \).

Therefore, the force of attraction is inversely proportional to the square of the distance between the two objects.

Comparing with Options

Let's look at how the force is inversely proportional to the different powers of distance presented in the options:

• Option 1: Inversely proportional to \( d^{1/2} \) (square root of distance). This would mean \( F \propto \frac{1}{d^{1/2}} \). This is not consistent with Newton's Law.
• Option 2: Inversely proportional to \( d^3 \) (cube of distance). This would mean \( F \propto \frac{1}{d^3} \). This is not consistent with Newton's Law.
• Option 3: Inversely proportional to \( d^2 \) (square of distance). This means \( F \propto \frac{1}{d^2} \). This is consistent with Newton's Law of Universal Gravitation.
• Option 4: Inversely proportional to \( d \) (distance). This would mean \( F \propto \frac{1}{d} \). This is not consistent with Newton's Law.

Based on Newton's Law, the force of attraction is inversely proportional to the square of the distance.

Revision Table: Gravitational Force and Distance

Additional Information: The Inverse Square Law

• The relationship where a physical quantity is inversely proportional to the square of the distance from the source is known as an inverse square law.
• Gravity follows an inverse square law.
• Other physical phenomena that follow an inverse square law include electric force between point charges (Coulomb's Law) and the intensity of light or sound from a point source.
• This inverse square relationship for gravity means that if you double the distance between two objects, the gravitational force between them becomes four times weaker (\( 1/(2d)^2 = 1/(4d^2) \)). If you triple the distance, the force becomes nine times weaker (\( 1/(3d)^2 = 1/(9d^2) \)).